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Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

  • Daniel M. Kane (a1) and Scott Duke Kominers (a2)
Abstract

For relatively prime positive integers u 0 and r, we consider the least common multiple L n := lcm(u 0, u 1..., u n ) of the finite arithmetic progression . We derive new lower bounds on L n that improve upon those obtained previously when either u 0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u 0 properly chosen, and is also nearly sharp as n → ∞.

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[BKS02] Bateman, P., Kalb, J., and Stenger, A., A limit involving least common multiples: 10797. Amer. Math. Monthly 109 (2002), no. 4, 393394. http://dx.doi.org/10.2307/2695513
[Far05] Farhi, B., Minorations non triviales du plus petit commun multiple de certaines suites finies d’entiers. C. R. Math. Acad. Sci. Paris 341 (2005), no. 8, 469474. http://dx.doi.org/10.1016/j.crma.2005.09.019
[Far07] Farhi, B., Nontrivial lower bounds for the least common multiple of some finite sequences of integers. J. Number Theory 125 (2007), no. 2, 393411. http://dx.doi.org/10.1016/j.jnt.2006.10.017
[FK09] Farhi, B. and Kane, D., New results on the least common multiple of consecutive integers. Proc. Amer. Math. Soc. 137 (2009), no. 6, 19331939. http://dx.doi.org/10.1090/S0002-9939-08-09730-X
[Han72] Hanson, D., On the product of the primes. Canad. Math. Bull. 15 (1972), 3337. http://dx.doi.org/10.4153/CMB-1972-007-7
[HF06] Hong, S. and Feng, W., Lower bounds for the least common multiple of finite arithmetic progressions. C. R. Acad. Sci. Paris 343 (2006), no. 1112, 695698. http://dx.doi.org/10.1016/j.crma.2006.11.002
[HK10] Hong, S. and Kominers, S. D., Further improvements of lower bounds for the least common multiples of arithmetic progressions. Proc. Amer. Math. Soc. 138 (2010), no. 3, 809813. http://dx.doi.org/10.1090/S0002-9939-09-10083-7
[HQ11] Hong, S. and Qian, G., The least common multiple of consecutive arithmetic progression terms. Proc. Edinb. Math. Soc. (2) 54 (2011), no. 2, 431441. http://dx.doi.org/10.1017/S0013091509000431
[HY08] Hong, S. and Yang, Y., Improvements of lower bounds for the least common multiple of finite arithmetic progressions. Proc. Amer. Math. Soc. 136 (2008), no. 12, 41114114. http://dx.doi.org/10.1090/S0002-9939-08-09565-8
[IK04] Iwaniec, H. and Kowalski, E., Analytic number theory. American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.
[Nai82] Nair, M., On Chebyshev-type inequalities for primes. Amer. Math. Monthly 89 (1982), no. 2, 126129. http://dx.doi.org/10.2307/2320934
[WTH13] Wu, R., Tan, Q., and Hong, S., New lower bounds for the least common multiples of arithmetic progressions. Chin. Ann. Math. Ser. B 34 (2013), no. 6, 861864. http://dx.doi.org/10.1007/s11401-013-0805-9
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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